Newform orbit 324.6.i.a

• Label: 324.6.i.a
• Level: $324$
• Weight: $6$
• Character orbit: 324.i
• Analytic conductor: $51.964$
• Analytic rank: $0$
• Dimension: $90$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	324.i (of order \(9\), degree \(6\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(51.9643576194\)
Analytic rank:	\(0\)
Dimension:	\(90\)
Relative dimension:	\(15\) over \(\Q(\zeta_{9})\)
Twist minimal:	no (minimal twist has level 108)
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 90 q + 87 q^{5}+O(q^{10}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
37.1		0			0			0		−88.2269	+	32.1120i		0		−34.2274	−	194.113i		0			0			0
37.2		0			0			0		−75.6746	+	27.5433i		0		0.725234	+	4.11300i		0			0			0
37.3		0			0			0		−70.4999	+	25.6599i		0		9.90571	+	56.1781i		0			0			0
37.4		0			0			0		−66.1386	+	24.0725i		0		36.2345	+	205.496i		0			0			0
37.5		0			0			0		−22.9538	+	8.35451i		0		4.55553	+	25.8357i		0			0			0
37.6		0			0			0		−16.6193	+	6.04891i		0		−21.0691	−	119.489i		0			0			0
37.7		0			0			0		−8.79199	+	3.20002i		0		35.7129	+	202.538i		0			0			0
37.8		0			0			0		−1.92729	+	0.701475i		0		−28.8149	−	163.417i		0			0			0
37.9		0			0			0		8.26716	−	3.00900i		0		−13.3174	−	75.5266i		0			0			0
37.10		0			0			0		38.2275	−	13.9137i		0		29.6162	+	167.962i		0			0			0
37.11		0			0			0		53.8777	−	19.6099i		0		−15.9538	−	90.4786i		0			0			0
37.12		0			0			0		59.0469	−	21.4913i		0		21.5227	+	122.061i		0			0			0
37.13		0			0			0		69.9871	−	25.4732i		0		9.15237	+	51.9057i		0			0			0
37.14		0			0			0		84.6372	−	30.8054i		0		−41.3474	−	234.493i		0			0			0
37.15		0			0			0		93.2376	−	33.9357i		0		7.30482	+	41.4277i		0			0			0
73.1		0			0			0		−82.0644	+	68.8602i		0		4.01162	−	1.46011i		0			0			0
73.2		0			0			0		−58.1486	+	48.7924i		0		−131.626	+	47.9080i		0			0			0
73.3		0			0			0		−50.4250	+	42.3116i		0		105.429	−	38.3730i		0			0			0
73.4		0			0			0		−40.6723	+	34.1282i		0		234.829	−	85.4708i		0			0			0
73.5		0			0			0		−25.7256	+	21.5863i		0		−179.693	+	65.4027i		0			0			0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
27.e	even	9	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	324.6.i.a		90
3.b	odd	2	1		108.6.i.a	✓	90
27.e	even	9	1	inner	324.6.i.a		90
27.f	odd	18	1		108.6.i.a	✓	90

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
108.6.i.a	✓	90	3.b	odd	2	1
108.6.i.a	✓	90	27.f	odd	18	1
324.6.i.a		90	1.a	even	1	1	trivial
324.6.i.a		90	27.e	even	9	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(324, [\chi])\).